Fig. 1.

Shining an input wave $|{\psi }_{In}\rangle$ on the linear optical object O leads to an output wave $|{\psi }_{Out}\rangle$ as a result of the linear interaction described by the operator or matrix S. If we shine one of the orthogonal input mode-converter basis function waves, such as the pth such (normalized) input mode $|{\psi }_{In\hspace{0.17em}p}\rangle ,$ then the resulting output wave is the corresponding pth orthogonal (normalized) output mode-converter basis function wave $|{\psi }_{Out\hspace{0.17em}p}\rangle$ but with (complex) amplitude s_p, where s_p is the corresponding pth singular value of S.